Space asymptotic method for the study of neutron propagation

Space asymptotic theory is shown to be a suitable model for the study of pulsed experiments in neutron multiplying systems. After a short revisitation of the basic aspects of space asymptotic theory applied on the Laplace transformed one-group transport equation, the full solution is derived. It is shown how results are exact in representing localized pulse propagation in the first portion of the transient, until the boundary is reached by the neutron signal, since it propagates with a finite velocity. Approximate models are then derived starting from the exact formulation and the B_N method is used to account for anisotropy effects. Numerical results are presented for one-dimensional systems, discussing the physical phenomena and noting the distortions introduced by approximate models, which may then turn out to be inadequate for the simulation of realistic pulsed experiments situations.     

Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem-I

We describe a number of methods for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally. For example, we consider an off-centre point source, a ring source and a disk source, or any combination of these, and calculate the surface scalar flux as a function of the radial and angular co-ordinates. Fourier transforms in the transverse directions are used and a Laplace transform in the axial direction. This enables the Wiener–Hopf method to be employed, followed by an inverse Fourier–Hankel transform. Some additional transformations are introduced which enable the inverse Hankel transforms involving Bessel functions to be evaluated numerically more efficiently. A hybrid diffusion theory method is also described which is shown to be a useful guide to the general behaviour of the solutions of the transport equation.     

基于先进节块法的六边形三维中子物理程序的开发与验证

六边形燃料组件在液态金属冷却快堆尤其是钠冷快堆中被广泛应用,针对这类堆型的设计与安全分析需要对堆芯中子通量与中子流进行三维全堆芯耦合计算。经过多年发展,目前已有多种解析节块法、积分节块法、节块展开法等先进节块法能在笛卡尔坐标系下较为精确求解多维中子扩散方程。本文通过径向半解析节块法耦合轴向高阶节块展开法的综合节块方法开发了反应堆三维中子物理计算软件SA HNHEX,并对VVER 440二维、三维基准题进行建模与仿真计算。计算结果与参考值符合较好,初步验证了使用该方法进行反应堆堆芯中子扩散计算的正确性。     

Effect of axial reflector on radial uniformity in neutron transmutation doping of silicon

Significant axial variation of radial uniformity is observed in Si-ingot neutron transmutation doping in the flux screening method, and leads to non-uniform resistivity distribution for a certain part of Si-ingot. This axial variation of radial uniformity is caused by the installation of a partial neutron screen which decreases the reaction rates differently in the center and surface at the region not surrounded by the partial neutron screen. For the improvement of the specific distribution of radial uniformity in the axial direction, a new concept of axial reflector is introduced to partly change the reaction rate at a certain region of Si-ingot, and neutron irradiation experiments are carried out at the heavy water neutron irradiation facility in the Kyoto University Research Reactor. Based on the experimental and numerical results, the new axial reflector is proved to be effective for improving the axial variation of radial uniformity.     

A finite fourier transform method for three dimensional steady state reactor core calculations

A method based on finite Fourier transform technique has been developed to solve the steady state multigroup neutron diffusion equation for reactor core calculations in X-Y/X-Y-Z geometry. In the present work, the neutron source in a node is approximated by a 5 point quadratic in X-Y plane and a quadratic in axial direction. The partial currents on the surfaces of the node have been assumed to be constant and equal to their average values. The equations cast in response matrix form are solved by standard fission source iterative approach. The outer iterations are accelerated using a coarse mesh rebalancing scheme. The algorithm has been implemented in a computer code finfor-sqr. The code has been validated by analysing a few benchmark problems.     

中子传输矩阵研究

通过简化假设,分析了中子传输矩阵的物理意义,推导出中子传输矩阵数学模型,并利用以往的数据进行了验证.同时根据矩阵的共轭梯度算法理论,研究利用堆外核探测器系统(RPN)的功率量程通道(PRC)6节电离室信号及堆内中子通量测量系统(RIC)获得的堆内通量分布信号计算中子传输矩阵的方法.这种算法得到的中子传输矩阵,可以植入冷却剂丧失(LOCA)监测系统(LSS系统);通过LSS系统可以实时监测堆芯轴向功率分布,进而监测堆芯轴向线功率密度.     

Two-dimensional nodal transport method for triangular geometry

The advanced nodal method for solving the multi-group neutron transport equation in two-dimensional triangular geometry is developed. To apply the transverse integration procedure, an arbitrary triangular node is transformed into a regular triangular node using coordinate transformation. The angular distributions of intra-node neutron fluxes and its transverse-leakage are represented by the S_N quadrature set. The spatial distributions of neutron flux and source in the regular triangle are given approximately by an orthogonal quadratic polynomial, and the spatial expansion of transverse-leakage is approximated by a second-order polynomial. To establish a stable and efficient iterative scheme, the improved nodal-equivalent finite difference algorithm is used. The results for several benchmark problems demonstrate the higher capability of the method to yield the accurate results in significantly smaller computing times than those required by the standard finite difference method and the finite element spherical-harmonics method.     

基于非结构网格的三维中子输运方程求解方法

本文使用离散纵标-间断有限元方法求解了三维中子输运方程,它对能量变量采用多群近似离散,对方向变量采用离散纵标法离散,对空间变量采用间断有限元离散;并研究了每个S_N离散方向的有限元网格的排序以及中子输运DFEM方程中几个矩阵的矩阵元的精确求解方法,并据此开发了基于非结构网格的三维输运计算程序TetTran1.0。基准例题校核结果表明,该程序具有很高的计算精度。     

Coarse-Mesh Three-Dimensional Transport Calculation Method

A coarse-mesh 3-D (X-Y-Z, Hexagonal-Z) discrete ordinates transport calculation method has been developed. This method employs an weighted diamond difference approximation, the weight in which is a function of neutron direction and scalar flux, and can be easily incorpolated into conventional discrete ordinates transport codes. Results obtained in four-group S₄P₀ calculations on simple fast reactors indicate that, though the computer time of this method has increased by 30–50% compared with that of the conventional finite difference method using the same mesh, the errors of k_eff and the power distribution are reduced remarkably.     

Miscellaneous remarks on choice of moments, moment reduction, local- and global-error bounds

Some observations are made on the merits of Legendre, half-range Legendre, Walsh and E_N representations for the angular dependence of neutron transport in finite element calculations. The splicing of transport and diffusion theory calculations is advocated to reduce the complexity of finite element calculations. To assess the accuracy of the splicing method it is necessary to know the error made by a finite element calculation in a local region of the system. This can be done with some precision by the method of bivariational bounds, which gives in a test example an error of for the captures in a locality.     

Spherical Harmonics Solutions of Multi-Dimensional Neutron Transport Equation by Finite Fourier Transformation

A method of solution of a monoenergetic neutron transport equation in P_L approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.     

用于求解细网SP3中子输运方程的两节块方法的精度与效率分析

采用两节块方法求解细网3阶简化球谐函数(SP3)中子输运方程,该方法只对零阶角通量密度的拉普拉斯算子进行节块法处理,对应的零阶通量密度采用2阶展开,横向泄漏采用零阶近似;以此方法开发了适用于细网全堆输运计算的CORCA-PIN程序,该程序同时集成了细网有限差分方法。验证算例采用KAIST 3A基准问题及扩展三维问题。数值结果表明,采用栅元1×1划分的两节块法具有可接受的计算精度,而计算时间只有相同精度的细网有限差分方法的11%。因此,本文提出的两节块方法适用于细网SP3中子输运方程计算。     

Soft computing systems applied to PWR''s xenon

The present work intends to introduce a soft computing technique as an effective and robust tool available to deal with nuclear engineering problems. This goal is reached by the presentation of an application: a genetic programming system (GP) able to automatically design a controller for the axial xenon oscillations in a pressurized water reactors (PWRs). The axial xenon oscillations control methodology is based on three axial offsets: the xenon axial offset (AOx), the iodine axial offset (AOi) and the neutron flux axial offset (AOf), effectively used in former work. Simulations were made using a two-point xenon oscillation model which employs the non-linear xenon and iodine balance equations and the one group, one-dimensional neutron diffusion equation, with non-linear power reactivity feedback, also proposed in the literature. Obtained results showed the ability of the GP in finding a strategy which can effectively control the axial xenon oscillations.     

Production and detection of fission-induced neutrons following fast neutron direct interrogation to various dry materials containing 235U

For better nuclear material accountancy, we had developed a non-destructive assay system dedicated to uranium waste drums (JAWAS-N: JAEA Waste Assay System at Ningyo-toge). The system is based on a fast neutron direct interrogation (FNDI) method. To clarify the characteristics of the FNDI method and the performance of JAWAS-N, experimental and computational mock-up tests were carried out using various dry materials that contained known amounts of natural uranium. As a result, linearity between the die-away time (τ₂) and the counts of fast neutrons attributed to ²³⁵U fission was confirmed. Moreover, the MCNP simulation was performed to discuss the radial and axial dependences of ²³⁵U fission probability, neutron detection efficiency, and sensitivity on uranium distributions in the drum. The simulation results agreed with the empirical results reported in a previous paper, providing valuable information on the practice of FNDI-based uranium determination. Furthermore, the nominal detection limits of natural uranium in JAWAS-N were estimated to be 15, 4, and 2 g for τ₂ = 0.2, 0.3, and 0.4 msec, respectively. The findings obtained here will contribute to the implementation of the FNDI method to assess the quantities of ²³⁵U in actual uranium waste drums.     

Numerical Solution to Space-Angle Energy-Dependent Neutron Integral Transport Equation

A numerical approach to the steady-state, space-, angle- and energy-dependent neutron transport equation is presented for neutron shielding calculations. The scattering integral, with anisotropic treatment of elastic scattering and isotropic treatment of inelastic scattering, is evaluated by the use of Gaussian and straightforward quadratures. A system of coupled one-group integral equations for all the energy meshes of interest, converted from the energy-dependent integral transport equation, is calculated by performing a line integration along the neutron path in the direction of motion. For this purpose the direction of neutron motion is represented by discrete-ordinate directions Ω_pq on the unit sphere.

The final presentation of the integral transport equation is derived in a difference form convenient for machine computation. A computation program PALLAS has been written in Fortran IV for IBM 360-75 computer to perform neutron transport calculations based on this approach.

Comparisons are given of the numerical solutions with analytical solutions for unscattered fluxes in various geometries such as plane, spherical and two-dimensional cylindrical, for volume sources with self absorption, and with experimental spectra for angular neutron fluxes in graphite-, polyethylene- and water-shield. Excellent agreement is obtained between the present calculations and analytical or experimental results.     

An analytical solution for the general perturbative diffusion equation by integral transform techniques

In this work, we present analytical solutions for the eigenvalue problem of a neutron flux in a rectangular two dimensional geometry by a two step integral transform procedure. For a given effective multiplication factor _Keff$K_{\text{eff}}$ we consider a homogeneous problem for two energy groups, i.e. fast and thermal neutrons, respectively, where the problem is setup by two coupled bi-dimensional diffusion equations in agreement with general perturbation theory (GPT). These are solved in a two-fold way by integral transforms, in the sequence Laplace transform followed by GITT and vice versa. Although, the functional base and the employed integral transforms are the same for both sequences, the procedures differ. We verify the efficiency of the sequence on the solutions of such problems, further the results are compared to the solution obtained by the finite difference method.     

A nodal SN transport method for three-dimensional triangular-z geometry

A new nodal S_N transport method has been developed to perform accurate transport calculation in three-dimensional triangular-z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S_N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal-z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k_eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.     

YOSHIO NISHINA CENTENNIAL SYMPOSIUM on EVOLUTIONAL TRENDS OF PHYSICAL SCIENCE

Axially simplified method of characteristics in three-dimensional geometry (ASMOC3D) has been proposed, and the 3D neutron transport calculation code based on this method, SHIKOKU, has been developed. Since ASMOC3D handles 3D problems by a two-dimensional (2D) neutron track set and simplified treatment in axial direction, the required memory and the computational time are expected to be less than those required by a direct 3D characteristics calculation scheme. SHIKOKU is applied to two problems of 3D geometry and the results of these problems show good agreements with the reference solutions obtained by a Monte-Carlo code. SHIKOKU is also applied to an actual three-loop-type PWR core. The prediction error of the radial power distribution is satisfactory and it is shown that the computational time and the required memory for a whole-core calculation by ASMOC3D are not prohibitive for presently available PCs.     

Generalization of New Finite Element Method Using ‘Imaginary’ Nodal Points

A method for improving the accuracy of finite element solutions to diffusion equations has been developed. The author previously suggested a method for improving the accuracy of finite element solutions to neutron diffusion equations, a kind of Helmholtz equations, within a short computing time. The method has been generalized so that it can be applied to problems described by the Laplace equation, too, such as temperature distributions and electric fields. In this generalized method, 3 ‘imaginary’ nodal points are added at the midsides of each data-given triangular element and the element is subdivided into 4 triangular subelements of the same dimension to improve accuracy. Then, approximate expressions, which express solutions at the ‘imaginary’ nodal points using those at ‘real’ nodal points, are derived by Jacobi's iteration method. These approximate expressions are used to reduce the number of unknowns in the final linear equations. The computing time required for the method described here is much shorter than that required for the straightforward method of increasing the number of elements 4 times under the same accuracy.     

基于钒探测器补偿的轴向偏移控制设计

机械补偿控制是基于控制棒调节反应堆功率并进行轴向功率偏移控制的先进技术,频繁移动的控制棒对传统堆外校准后的轴向功率偏差测量精度有较大的负面影响,采用能给出准确堆内功率分布的钒基自给能探测器的信号可对其进行校正。为补偿钒探测器较长的响应时间,设计了超前/滞后控制器,提出了用补偿后的信号对堆外测量得到的轴向功率偏差信号进行修正的方法。仿真结果表明,该方法能有效应对控制棒移动对轴向偏移控制的影响,可提高控制精度。对该方法的安全相关影响、可实施性及性能分析表明,其具有较高的实用价值。